The focus of our work is centered around the fact that reformulations
of the spectral method in the \(p\) -norm, for \(p \in (1,2)\) , lead to a sharp approximation of balanced cut
metrics [1]}, [2]}. This variant results in a tight relaxation of the spectral clustering problem, with the resulting solutions approximating closely the solution of the original discrete problem. Partitions are obtained by thresholding the eigenvector associated with the second-smallest eigenvalue of the graph \(p\) -Laplacian, a nonlinear generalization of the graph Laplacian. It has been proven in  [3]} that the resulting cut converges to the optimal Cheeger cut [4]} for \(p \rightarrow 1\) , thus demonstrating the superiority of \(p\) -spectral clustering over its traditional 2-norm counterpart.
